Cercignani Slow Rarefied Flows

Theory and Application to Micro-Electro-Mechanical Systems
2006
ISBN: 978-3-7643-7537-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 41, 166 Seiten

Reihe: Progress in Mathematical Physics

This volume is intended to coverthe presentstatus of the mathematicaltools used to deal with problems related to slow rare?ed ?ows. The meaning and usefulness of the subject, and the extent to which it is covered in the book, are discussed in some detail in the introduction. In short, I tried to present the basic concepts and the techniques used in probing mathematical questions and problems which arise when studying slow rare?ed ?ows in environmental sciences and micromachines. For the book to be up-to-date without being excessively large, it was necessary to omit some topics, which are treated elsewhere, as indicated in the introd- tion and, whenever the need arises, in the various chapters of this volume. Their omission does not alter the aim of the book, to provide an understanding of the essential mathematical tools required to deal with slow rare?ed ?ows and give the background for a study of the original literature. Although I have tried to give a rather complete bibliographical coverage,the choice of the topics and of the references certainly re?ects a personal bias and I apologize in advance for any omission. I wish to thank Lorenzo Valdettaro, Antonella Abb` a, Silva Lorenzani and Paolo Barbante for their help with pictures and especially Professor Ching Shen for his permission to reproduce his pictures on microchannel ?ows.

Autoren/Hrsg.

Cercignani, Carlo

Inhaltsverzeichnis

1;Contents;5
2;Preface;7
3;Introduction;8
4;1 The Boltzmann Equation;11
4.1;1.1 Historical Introduction;11
4.2;1.2 The Boltzmann Equation;14
4.3;1.3 Molecules Di.erent from Hard Spheres;21
4.4;1.4 Collision Invariants;22
4.5;1.5 The Boltzmann Inequality and the Maxwell;25
4.6;Distributions;25
4.7;1.6 The Macroscopic Balance Equations;26
4.8;1.7 The;30
4.9;theorem;30
4.10;1.8 Equilibrium States and Maxwellian Distributions;32
4.11;1.9 The Boltzmann Equation in General Coordinates;34
4.12;1.10 Mean Free Path;35
4.13;References;36
5;2 Validity and Existence;39
5.1;2.1 Introductory Remarks;39
5.2;2.2 Lanford’s Theorem;40
5.3;2.3 Existence and Uniqueness Results;46
5.4;2.4 Remarks on the Mathematical Theory of the;49
5.5;Boltzmann Equation;49
5.6;References;49
6;3 Perturbations of Equilibria;50
6.1;3.1 The Linearized Collision Operator;50
6.2;3.2 The Basic Properties of the Linearized Collision;52
6.3;Operator;52
6.4;3.3 Some Spectral Properties;59
6.5;3.4 The Asymptotic Behavior of the Solution of the;69
6.6;Cauchy Problem for the Linearized Boltzmann;69
6.7;Equation;69
6.8;3.5 The Global Existence Theorem for the Nonlinear;72
6.9;Equation;72
6.10;3.6 Extensions: The Periodic Case and Problems in One;74
6.11;and Two Dimensions;74
6.12;References;75
7;4 Boundary Value Problems;77
7.1;4.1 Boundary Conditions;77
7.2;4.2 Initial-Boundary and Boundary Value Problems;82
7.3;4.3 Properties of the Free-streaming Operator;89
7.4;4.4 Existence in a Vessel with an Isothermal Boundary;92
7.5;4.5 The Results of Arkeryd and Maslova;93
7.6;4.6 Rigorous Proof of the Approach to Equilibrium;96
7.7;4.7 Perturbations of Equilibria;98
7.8;4.8 A Steady Flow Problem;99
7.9;4.9 Stability of the Steady Flow Past an Obstacle;105
7.10;4.10 Concluding Remarks;107
7.11;References;108
8;5 Slow Flows in a Slab;111
8.1;5.1 Solving the Linearized Boltzmann Equation in a;111
8.2;Slab;111
8.3;5.2 Model Equations;117
8.4;5.3 Linearized Collision Models;119
8.5;5.4 Transformation of Models into Pure Integral;121
8.6;Equations;121
8.7;5.5 Variational Methods;123
8.8;5.6 Poiseuille Flow;131
8.9;References;136
9;6 Flows in More Than One Dimension;138
9.1;6.1 Introduction;138
9.2;6.2 Linearized Steady Problems;138
9.3;6.3 Linearized Solutions of Internal Problems;143
9.4;6.4 External Problems;146
9.5;6.5 The Stokes Paradox in Kinetic Theory;147
9.6;References;150
10;7 Rarefied Lubrication in Mems;152
10.1;7.1 Introductory Remarks;152
10.2;7.2 The Modi.ed Reynolds Equation;153
10.3;7.3 The Reynolds Equation and the Flow in a;156
10.4;Microchannel;156
10.5;7.4 The Poiseuille-Couette Problem;158
10.6;7.5 The Generalized Reynolds Equation for Unequal;163
10.7;Walls;163
10.8;7.6 Concluding remarks;167
10.9;References;168
11;Index;171